While studying differential geometry, I came across this problem.
When you try to find the metric or the basis vectors of the 2-sphere, you can use the transformation law.
$$e'_i = \frac{dx^j}{dx'^i}e_j$$
And the same with the metric tensor.
However, I can not do this with the Riemann tensor or the Ricci tensor because the not-curved manifold where we have our 2-sphere has vanishing Riemann tensors. In that case, we can not use the transformation rule.
So, when can we use the transformation rule with submanifolds?